I've written about the options for reducing this effect; one is to use some kind of anti-shake mount. These range in price and complexity from simple rubber insulators through to expensive brushless mounts.
I tried a simple anti-shake mount, the Anti Vibration Anti-Jello Vibration Isolator Low Profile Carbon Fiber Mount (below) and it does seem to make a difference.
The two clips below show footage taken using the anti-shake mount. The first sequence is the "raw" footage; it still has some shaky moments, but that can mostly be removed using Premiere Pro/s Warp Stabilizer, as can be seen in the second clip. Note: the footage was shot at dusk.
The following footage has the Premiere Pro Warp Stabilizer applied, and has been graded using Red Giant Looks to give it more of a film look.
Prop BalancerThe other suggestion for reducing the shake is to do 'prop-balancing.' The theory is that if the prop isn't balanced, it can cause vibration.
I bought a prop balancer - which cost less than $10 - and gave it a go...and pretty quickly decided against doing anything to the prop a few minutes of experimentation.
The problem with prop balancing, at least with the Phantom's props, is that the mount hole isn't perfectly circular on one side, and it's practically impossible to determine whether you have actually got the center of the prop when you attach it to the prop balancer. And if you don't have it centered, your balancing is going to be pointless.
I tightened the prop between the two conical nuts that are used to hold the prop and then put it in the rig. The prop appeared to be slightly off-balance. BUT, releasing the prop and then reattaching it (trying to get it "centered" both times) I discovered that it was just as likely to register that the other side was heavier.
Perhaps a more expensive prop-balancer would do a more accurate job, though I'm not sure how that would work unless the prop balancer used an attachment point that was the same as the post on the Phantom itself.
Either way, I think the prop is pretty closely balanced, so I'm not going to alter them.